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If the mean and variance of four numbers 3, 7, x and y(x > y) be 5 and 10 respectively, then the mean of four numbers 3 + 2x, 7 + 2y, x + y and x - y is ______.
Correct answer is '12'. Can you explain this answer?
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If the mean and variance of four numbers 3, 7, x and y(x > y) be 5 ...
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If the mean and variance of four numbers 3, 7, x and y(x > y) be 5 ...
To find the value of x and y, we need to use the formulas for mean and variance.
Mean (μ) is calculated by summing all the numbers and dividing by the total count:
μ = (3 + 7 + x + y) / 4
Variance (σ^2) is calculated by finding the average of the squared differences between each number and the mean:
σ^2 = [(3-μ)^2 + (7-μ)^2 + (x-μ)^2 + (y-μ)^2] / 4
Given that the mean is 5 and the variance is 4, we can substitute these values into the formulas:
5 = (3 + 7 + x + y) / 4
4 = [(3-5)^2 + (7-5)^2 + (x-5)^2 + (y-5)^2] / 4
Simplifying the equations:
20 = 3 + 7 + x + y
16 = (3-5)^2 + (7-5)^2 + (x-5)^2 + (y-5)^2
20 = 10 + x + y
16 = 4 + (x-5)^2 + (y-5)^2
Combining like terms:
x + y = 10
(x-5)^2 + (y-5)^2 = 12
From the first equation, we can solve for x in terms of y:
x = 10 - y
Substituting this into the second equation:
(10-y-5)^2 + (y-5)^2 = 12
(5-y)^2 + (y-5)^2 = 12
Expanding and simplifying:
y^2 - 10y + 25 + y^2 - 10y + 25 = 12
2y^2 - 20y + 50 = 12
2y^2 - 20y + 38 = 0
Dividing by 2:
y^2 - 10y + 19 = 0
Using the quadratic formula:
y = (10 ± √(10^2 - 4*1*19)) / (2*1)
y = (10 ± √(100 - 76)) / 2
y = (10 ± √24) / 2
y = (10 ± 2√6) / 2
Simplifying:
y = 5 ± √6
Therefore, the possible values for y are 5 + √6 and 5 - √6.
Substituting these values back into the equation x = 10 - y:
If y = 5 + √6, then x = 10 - (5 + √6) = 5 - √6
If y = 5 - √6, then x = 10 - (5 - √6) = 5 + √6
So the possible values for x are 5 - √6 and 5 + √6.
Mean (μ) is calculated by summing all the numbers and dividing by the total count:
μ = (3 + 7 + x + y) / 4
Variance (σ^2) is calculated by finding the average of the squared differences between each number and the mean:
σ^2 = [(3-μ)^2 + (7-μ)^2 + (x-μ)^2 + (y-μ)^2] / 4
Given that the mean is 5 and the variance is 4, we can substitute these values into the formulas:
5 = (3 + 7 + x + y) / 4
4 = [(3-5)^2 + (7-5)^2 + (x-5)^2 + (y-5)^2] / 4
Simplifying the equations:
20 = 3 + 7 + x + y
16 = (3-5)^2 + (7-5)^2 + (x-5)^2 + (y-5)^2
20 = 10 + x + y
16 = 4 + (x-5)^2 + (y-5)^2
Combining like terms:
x + y = 10
(x-5)^2 + (y-5)^2 = 12
From the first equation, we can solve for x in terms of y:
x = 10 - y
Substituting this into the second equation:
(10-y-5)^2 + (y-5)^2 = 12
(5-y)^2 + (y-5)^2 = 12
Expanding and simplifying:
y^2 - 10y + 25 + y^2 - 10y + 25 = 12
2y^2 - 20y + 50 = 12
2y^2 - 20y + 38 = 0
Dividing by 2:
y^2 - 10y + 19 = 0
Using the quadratic formula:
y = (10 ± √(10^2 - 4*1*19)) / (2*1)
y = (10 ± √(100 - 76)) / 2
y = (10 ± √24) / 2
y = (10 ± 2√6) / 2
Simplifying:
y = 5 ± √6
Therefore, the possible values for y are 5 + √6 and 5 - √6.
Substituting these values back into the equation x = 10 - y:
If y = 5 + √6, then x = 10 - (5 + √6) = 5 - √6
If y = 5 - √6, then x = 10 - (5 - √6) = 5 + √6
So the possible values for x are 5 - √6 and 5 + √6.
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If the mean and variance of four numbers 3, 7, x and y(x > y) be 5 and 10 respectively, then the mean of four numbers 3 + 2x, 7 + 2y, x + y and x - y is ______.Correct answer is '12'. Can you explain this answer?
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If the mean and variance of four numbers 3, 7, x and y(x > y) be 5 and 10 respectively, then the mean of four numbers 3 + 2x, 7 + 2y, x + y and x - y is ______.Correct answer is '12'. Can you explain this answer? for JEE 2025 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about If the mean and variance of four numbers 3, 7, x and y(x > y) be 5 and 10 respectively, then the mean of four numbers 3 + 2x, 7 + 2y, x + y and x - y is ______.Correct answer is '12'. Can you explain this answer? covers all topics & solutions for JEE 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If the mean and variance of four numbers 3, 7, x and y(x > y) be 5 and 10 respectively, then the mean of four numbers 3 + 2x, 7 + 2y, x + y and x - y is ______.Correct answer is '12'. Can you explain this answer?.
If the mean and variance of four numbers 3, 7, x and y(x > y) be 5 and 10 respectively, then the mean of four numbers 3 + 2x, 7 + 2y, x + y and x - y is ______.Correct answer is '12'. Can you explain this answer? for JEE 2025 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about If the mean and variance of four numbers 3, 7, x and y(x > y) be 5 and 10 respectively, then the mean of four numbers 3 + 2x, 7 + 2y, x + y and x - y is ______.Correct answer is '12'. Can you explain this answer? covers all topics & solutions for JEE 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If the mean and variance of four numbers 3, 7, x and y(x > y) be 5 and 10 respectively, then the mean of four numbers 3 + 2x, 7 + 2y, x + y and x - y is ______.Correct answer is '12'. Can you explain this answer?.
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